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2.4 Factors Numbers that are multiplied together are called factors. Factors of a number, a, are numbers that when multiplied together produce a product of a. The number 12 has 6 possible factors: 1 x 12 = 12 2 x 6 = 12 3 x 4 = 12 So the factors are 1, 2, 3, 4, 6, and 12. A factor pair table can be written for 12 to find all the factors. Just start from 1 and keep incrementing and checking if 12 is divisible by it. When a factor you choose has already appeared in the table. You know you are finished. 12 1 12 2 6 3 4 4 3 3 was already in the table. Note that a number is always a factor of itself because a x 1 = a When two whole numbers, m and n, multiply to get a product, p (m x n = p), then we can say these facts: 1) p is a multiple of m and n 2) p is divisble by m and n 3) m and n divide evenly into p. 4) m and n are factors of p. Example: 12 is a multiple of 4 and a multiple of 3 12 is divisible by 3 and 4 3 and 4 divide evenly into 12 3 and 4 are factors of 12. Prime Numbers A prime number is a whole number, greater than 1, that has only 1 an itself as factors. Composite Numbers A composite number is a whole number, greater than 1, that are not prime. Prime Factorization To find the prime factorization of a whole number means to write it as the product of only prime numbers. This is can be useful when finding things like the Greatest Common Factor or Least Common Multiple between two numbers (we’ll get into that later). Example: Factor 90 into its prime factors. Choose any two factors of 90 (besides 1 and 90) Then do the same with each of those factors. Keep going until you have only prime factors as the bottom “roots” of the “factor tree.” 90 9 3 10 3 2 5 90 = 3 3 2 5 Putting these factors in numerical order and then combing like terms into exponents gives: 90 = 2 32 5 Theorem: Any composite number has exactly one set of prime factors. Example 5 Find the prime factorization of 210 First, pick any two factors of 210. For instance 21 and 10. We could have also picked 7 and 30 as the factors. 210 21 3 210 10 7 2 7 30 5 Notice that either method gives us 210 = 2 3 5 7 6 3 5 2 Alternate Factoring Method – I call it the “Pyramid Division Method” For more info: http://www.purplemath.com/modules/factnumb.htm With this method, you start and the bottom of the pyramid and move up, so you have to leave lots of room at the top of your problem. Example: Find the prime factorization of 90. Start with the first prime factor you can think of that goes into 90. We can’t choose 9 or 10 because they aren’t prime. Since 90 is even, start with the factor 2. 3 39 5 45 is 3 prime? Yes! You are done. is 9 prime? No. Keep dividing. is 45 prime? No. Keep dividing. 2 90 Prime Factorization of 90 is 2x5x3x3 Write the factors in numerical order and use exponents where factors are repeated = 2 32 5 Upside-Down Division Method Example: Find the prime factorization of 60 2 60 3 30 Is 30 prime? No. 2 10 Is 10 prime? No. 5 Is 5 prime? Yes. Stop Prime Factorization of 60 is 22 3 5